Geometry and Measurement: Ages Six to Nine
Jump to section:
2.GM.010 History of geometry
2.GM.020 History of Measurement
2.GM.030 Solid Geometry
Plane Geometry
2.GM.040 Plane Geometry: introduction
2.GM.050 Plane Geometry: constructive triangles
2.GM.060 Plane Geometry: fundamental concepts
2.GM.070 Plane Geometry: lines
2.GM.080 Plane Geometry: angles
2.GM.090 Plane Geometry: measurement of angles
2.GM.100 Plane Geometry: polygons
2.GM.110 Tessellation
2.GM.120 Symmetry
2.GM.130 Similarity, Congruence and Equivalence
Geometry Overview
The word geometry is derived from the Greek words for earth and measurement. The word geometry literally means ‘measuring the earth’. The Montessori approach to teaching geometry involves the use of concrete materials, which engage with the children’s sense of wonder and which encourage children to measure for themselves the world they live in.
In the Montessori curriculum the first knowledge about shapes and space is gained through activity with concrete materials in the Children’s House. This sensory work is extended in the geometry lessons in the environment prepared for children from six to nine years old. All new concepts continue to be introduced through activities in which children manipulate concrete materials, but from the age of six children independently complete a sequence of activities that build deeper understanding and move them towards abstraction. Manipulative work with concrete materials always precedes the presentation of an abstract concept or formula. Ideally, children, during their own independent work, begin to verbalise the concepts and formulae in conversation, although if, unusually, children do not make this step for themselves, an adult will model how to talk about the concept using the language of geometry.
Learning the names of geometric shapes and their parts is an important aspect of the study of geometry. As in the Children’s House children take part in spoken language lessons in which they learn the names of geometric shapes and their parts, but for children beyond the age of six, naming lessons also include:
learning the origin, or etymology, of the name
learning definitions and systems of classification
reading and writing the names.
In addition to the names, children of this age want to know the reasons for things so they are given opportunities to explore questions such as:
Why do we say this is an acute angle?
How many right angles would make a whole angle?
Following most presentations, children take part in follow-up activities with card materials and booklets.
The lessons in geometry are keys to enable children to follow their own interests in this area of study. They are encouraged to explore the concrete materials to find new relationships, variations and extensions. As in all areas of the Montessori curriculum for children of this age, the initial lessons and the children’s use of the concrete materials provides them with the knowledge they need as a starting point for their own exploration and discovery.
The study of geometry for children from the age of six begins with an exploration of the foundation concepts: point – line – surface – solid. These concepts are then explored in more detail in a sequence of follow-up lessons. From the same starting point, the actual sequence may vary from child to child. For example, work on lines and angles might be presented parallel to one another, and some early polygon work can be presented parallel to later work with angles. Whatever sequence is followed, work in each area progresses logically, with each new lesson building on the one before. For example, children complete work that builds an understanding of equivalence, before they work with area, and the work with area precedes the work on volume. In the case of the study of circles, the work follows a progression such as the following:
identification of linear, then surface parts
relationships between a line and a circle
relationships between two circles
relationships between circles and polygons
circumference, radius and their relationship
area of a circle
As in all areas of the Montessori Cosmic Education curriculum, the study of geometry is designed to encourage activity, repetition, exploration, discovery, logical thinking and reasoning. The initial lessons are clear and succinct, providing children with enough knowledge to allow them to ask pertinent questions, and enough guidance to initiate independent activity with the materials. It is through this activity that children make the knowledge their own and learn to research and think for themselves.
The study of measurement in the Montessori Cosmic Education curriculum, begins with an account of the historical development of measuring, comparing and estimating. The children explore the reasons humans in earlier times might have wanted to, or needed to, measure, compare and estimate. They also experiment with the use of non-standard units and units used in earlier times, in particular, units based on parts of the body, including, for example, the hand, the palm and the cubit. They use these units to measure objects in the environment. During these activities children recognise the need for standardised units, as exemplified, for example, in the story of the royal cubit from Ancient Egypt. They are then introduced to the standard measurement systems in use today, beginning with the International Metric System. As students learn different elements of this system, they apply their knowledge, first, to measuring activities in the everyday environment, selecting the most appropriate standard unit and recording the measurement of, for example, the length of the school hall, the temperature of the classroom, the volume of water in a watering can or the mass of a can of pet food. Finally, students apply their knowledge during activities and research projects across all areas of the curriculum and in the solving of real-life problems, including, for example, calculating how much water the garden needs each day, or how many days the bird food will last before more needs to be purchased.
History of Geometry 2.GM.010
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore the history of geometry and geometry in early civilisations
.02 Research use of geometry in different cultures and across history
＋ Materials and Activities
Activities include:
- the story of geometry
- independent research
- creating charts, models and timelines
- researching the use of geometry in earlier times e.g., Ancient Egypt, Mesopotamia
- researching the use of geometry across cultures e.g., Aboriginal and Torres Strait Islander cultures in Australia.
Resources include:
- charts, card material and artefacts
- reference and research materials (paper-based, digital, web-based, multimedia).
History of Measurement 2.GM.020
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore the history and purpose of measurement
.02 Explore and experiment with non-standard and standard units of measurement
＋ Materials and Activities
Activities include:
- stories and discussions about measurement
- brainstorming all the possible phenomena we can measure and ways we can measure
- activities to draw attention to the importance of standard units of measurement
- activities to become familiar with International Metric System
- experimenting with a range of units of measurement e.g., invented units; units used in the past, units used in other countries e.g., imperial units
- imagining how humans might measure things in the future
- creating timelines, charts and booklets.
Resources include:
- timeline of measurement
- charts, card material and artefacts
- reference and research materials (paper-based, digital, web-based, multimedia).
＋ Links to Australian Curriculum (ACv9)
Solid Geometry 2.GM.030
Knowledge, Skills and Understanding
Typically, children will:
.01 Introduce, review and/or consolidate knowledge of geometric solids
.02 Recognise, name and label geometric solids
.03 Sort and classify solids
.04 Develop and extend spatial awareness
＋ Materials and Activities
Activities include:
- demonstrations and lessons with wooden geometric solids to review names of geometric shapes, and to introduce etymology of names
- activities with Montessori geometric solids and manipulable bases
- labelling geometric solids (spoken and written)
- game of relationships
- describing and classifying solids e.g., according to base/surfaces
- model-making and technical drawing
- researching solids in the environment.
- building complex solid constructions using the Box of 250 Cubes from Cube Up! card materials
- assembling flatpack furniture for classroom from included instructions (e.g., IKEA)
- designing and building own furniture in teams from cardboard sheets and masking tape
- designing and creating Rube Goldberg machines
Resources include:
- basket of wooden geometry solids and manipulable bases
- labels and booklet
- construction equipment e.g., modelling clay, construction paper
- solids in the environment
- command cards
- CubeUp! card material from Nienhuis
- reference and research materials (paper-based, digital, web-based, multimedia).
＋ Links to Australian Curriculum (ACv9)
AC9MFSP01
AC9M1SP01
AC9M2SP01
AC9M3SP01
AC9MFSP02
AC9M1SP02
Plane Geometry: introduction 2.GM.040
Knowledge, Skills and Understanding
Typically, children will:
01 Review, name and consolidate knowledge of plane geometric shapes
＋ Materials and Activities
Activities include:
- demonstrations and lessons with geometry cabinet to review names of geometric shapes and to introduce etymology of names and properties of:
- triangle, square, circle
- types of triangles (scalene, isosceles, equilateral; acute-angled, obtuse-angled, equilateral)
- types of quadrilaterals (common quadrilateral, rectangle, square, parallelogram, rhombus, kite, trapezium)
- regular polygons (pentagon, hexagon, heptagon, octagon, nonagon, decagon)
- curved figures (circle, ellipse, oval)
- compound and curvilinear figures (curvilinear triangle, ‘flowers’)
- sorting shapes and replacing in frames
- labelling shapes
- tracing, cutting, gluing, drawing and writing activities
- drawing geometric shapes
- making charts and booklets
- researching shapes in the environment
- using a digital camera to photograph shapes, lines or angles for a photobook
Resources include:
- geometry cabinet
- shapes in the environment
- scissors, coloured paper and glue
- technical drawing equipment
- command cards
- reference and research materials (paper-based, digital, web-based, multimedia)
- digital camera
＋ Links to Australian Curriculum (ACv9)
AC9MFSP01
Plane Geometry: constructive triangles 2.GM.050
Knowledge, Skills and Understanding
Typically, children will:
.01 Use wooden triangles of different types to construct a variety of shapes
.02 Recognise and name constructed shapes
.03 Draw constructed shapes
＋ Materials and Activities
Activities include:
- using pairs of triangles to construct a variety of triangles and quadrilaterals
- discovering more shapes by sliding, pivoting and flipping triangles
- using sets of triangles to construct hexagons, pinwheels and related complex shapes
- tracing, cutting, gluing, drawing and labelling activities
- research in the classroom environment; including photographing shapes with a digital camera
Resources include:
- constructive triangles (rectangular boxes and triangular boxes)
- command cards
- scissors, coloured paper and glue
- technical drawing equipment
- command cards
- reference and research materials (paper-based, digital, web-based, multimedia
- digital camera
＋ Links to Australian Curriculum (ACv9)
AC9M1SP01
AC9M1SP01
Plane Geometry: fundamental concepts 2.GM.060
Knowledge, Skills and Understanding
Typically, children will:
.01 Build knowledge of fundamental geometry concepts:
point
line
surface
solid
＋ Materials and Activities
Activities include:
- games and activities for exploring concepts with everyday objects and concrete materials
- linking geometry (point, line, surface, solid) with concrete mathematics materials (unit bead, ten-bar, hundred-square, thousand-cube)
- labelling
- creating charts, booklets and models
- research in environment.
Resources include:
- everyday objects
- golden bead material
- models, card material and labels
- scissors, coloured paper and glue
- technical drawing equipment
- command cards
- reference and research materials (paper-based, digital, web-based, multimedia).
Plane Geometry: lines 2.GM.070
Knowledge, Skills and Understanding
Typically, children will:
.01 Identify types of lines: straight, curved
.02 Identify parts of lines: origin, ray
.03 Identify positions of a straight line: horizontal, vertical, oblique
.04 Identify relative positions of two straight lines: parallel, convergent, divergent, transversal, perpendicular
.05 Build definitions related to lines
＋ Materials and Activities
Activities include:
- demonstrations, stories, lessons and games, including lessons to introduce etymology of names
- manipulating and labelling models and pictures
- introduction to the box of sticks
- building and reading definitions
- drawing activities
- creating charts, booklets and models
- research lines in environment.
Resources include:
- box of sticks (colour-coded and calibrated sticks and arcs used to construct plane geometric figures)
- picture, label, definition and booklet material
- command cards
- drawing equipment
- reference and research materials (paper-based, digital, web-based, multimedia).
Plane Geometry: angles 2.GM.080
Knowledge, Skills and Understanding
Typically, children will:
.01 Identify parts of angles: vertex, arm
.02 Identify types of angles: acute, obtuse, whole
.03 Build definitions related to angles
＋ Materials and Activities
Activities include:
- demonstrations and activities using concrete materials to construct and label parts of angles and to build different types of angles
- learning the etymology of the terms
- labelling the parts of angles and types of angles on models and pictures
- building and reading definitions
- drawing and labelling activities
- creating charts and booklets
- researching angles in the environment.
Resources include:
- box of geometry sticks
- geometry charts
- drawing equipment
- envelopes with pre-cut paper figures (triangles, quadrilaterals, figures with five or more sides)
- picture, label, definition and booklet material
- command cards
- researching angles in the
- protractor, compass and ruler
- reference and research materials (paper-based, digital, web-based, multimedia).
＋ Links to Australian Curriculum (ACv9)
AC9M4M04
AC9M3M05
Plane Geometry: measurement of angles 2.GM.090
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand how to measure the size of an angle i.e., amount of turn
.02 Use unit of measurement for angles: degrees
.03 Measure angles and record measurements, using protractors
.04 Explore different types of angles: whole, convex, reflex
.05 Apply knowledge of addition and subtraction to angles
.06 Explore relations between the angles of polygons
＋ Materials and Activities
Activities include:
- story and demonstrations to introduce the measurement of angles: story of the star; the Babylonian civilisation
- demonstrations and activities for measuring angles
- tracing and measuring the angles of insets e.g., geometry cabinet, small insets
- using measurements to identify different types of angles
- creating and measuring own angles
- demonstrations and activities with concrete materials for operations with angles (addition, subtraction)
- bisecting angles using a compass
- researching the angles in regular and irregular polygons
- measuring angles in the environment
- making charts and booklets
- applying knowledge to real life tasks e.g., construction projects, model-making, design
- research activities e.g., how architects measure angles.
Resources include:
- Montessori protractor
- commercial protractors
- fraction insets, geometry cabinet insets, small insets
- command cards; word problems
- reference and research materials (paper-based, digital, web-based, multimedia).
＋ Links to Australian Curriculum (ACv9)
AC9M2M05
AC9M5M04
AC9M3M05
AC9M5M04
AC9M3M05
Plane Geometry: polygons 2.GM.100
Knowledge, Skills and Understanding
Typically, children will:
.01 Classify types of plane figures: closed curves-polygons, concave-convex
.02 Classify types of polygons: irregular, regular
.03 Classify polygons:
the seven triangles of reality
the seven quadrilaterals of reality
polygons with more than four sides
.04 Identify and label the parts of polygons: surface, perimeter, sides, angles, vertices, base, height, diagonal
＋ Materials and Activities
Activities include:
- demonstrations, stories and lessons to introduce each family of shapes e.g., the Story of Pythagoras to introduce the right-angled triangle
- constructing, manipulating, comparing and labelling plane figures using concrete material
- labelling images
- building and reading definitions
- drawing and labelling activities
- creating charts and booklets
- research plane figures in the environment.
Resources include:
- box of geometry sticks
- geometry charts
- drawing equipment
- picture, label, definition and booklet material
- command cards
- protractor, compass and ruler
- reference and research materials (paper-based, digital, web-based, multimedia).
Tessellation 2.GM.110
Knowledge, Skills and Understanding
Typically, children will:
.01 Tessellate figures
.02 Use knowledge of angles and their measurement to explore the mathematics of tessellation
＋ Materials and Activities
Activities include:
- demonstrations and games with concrete material to explore tessellation: which tiles? how many ways?, combining shapes, interstices
- command cards
- design activities (tracing, cutting, gluing, drawing, constructing, creating mosaics)
- activities involving the measurement and addition of angles
- researching tessellation in the environment
- research projects e.g., tessellation in nature, use of tessellation for building and decoration across time.
Resources include:
- tiling game (Box 1 and 2)
- Tangram activities
- command cards
- Montessori protractor
- reference and research materials (paper-based, digital, web-based, multimedia).
Symmetry 2.GM.120
Knowledge, Skills and Understanding
Typically, children will:
.01 Explore and understand the concept of symmetry
.02 Experiment with the axis of symmetry in a variety of shapes
＋ Materials and Activities
Activities include:
- demonstrations and games with concrete material
- exploring etymology of the terms e.g., symmetry, axis
- paper-folding, construction, drawing, labelling and design activities
- research in the environment
- applying the concept of symmetry e.g., in visual arts.
Resources include:
- constructive triangles
- coloured paper, glue
- drawing and visual arts equipment and resources
- command cards
- reference and research materials (paper-based, digital, web-based, multimedia).
Similarity, congruence and equivalence 2.GM.130
Knowledge, Skills and Understanding
Typically, children will:
.01 Identify, name and construct congruent, similar and equivalent figures
＋ Materials and Activities
Activities include:
- demonstrations, lessons and games with concrete material
- exploring etymology of terms and working with symbols
- paper-folding, construction, drawing, labelling and design activities
- making charts and booklets
- research in the environment
- applying concepts e.g., in visual arts, construction
- research projects.
Resources include:
- iron (small) insets
- constructive triangles (triangular box, large and small hexagonal boxes)
- insets of equivalence
- command cards
- objects in environment
- reference and research materials (paper-based, digital, web-based, multimedia).
Area of a Surface: introduction 2.GM.140
Knowledge, Skills and Understanding
Typically, children will:
.01 Become familiar with the concept of surface area
.02 Explore units of measurement for area
.03 Measure and record area using a grid
＋ Materials and Activities
Activities include:
- informal measurement activities e.g., make a square metre out of paper and use it to measure the area of the classroom
- comparing areas of familiar spaces e.g., rooms, playground
- using square grids to measure and compare the areas of surfaces e.g., in drawing, construction and mapping activities
- applying knowledge to everyday tasks e.g., construction and design projects
- simple exercises in counting squares to calculate area.
Resources include:
- everyday materials
- command cards; word problems
- reference and research materials (paper-based, digital, web-based, multimedia).
Volume: liquid capacity 2.GM.150
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand and apply the concept of liquid capacity (volume)
.02 Learn units of measurement for liquid capacity (litres, millilitres)
.03 Measure and record liquid capacity
.04 Choose the most appropriate unit of liquid capacity for amount to be measured
.05 Research other measurements of liquid capacity
＋ Materials and Activities
Activities include:
- filling and pouring activities to compare the liquid capacity of familiar containers
- activities to become familiar with units for measuring liquid capacity including history of the litre, relationship between litre and smaller units (millilitre) and larger units (megalitre), etymology and meaning of the word metre, prefixes (milli-, mega-), abbreviations
- measuring the liquid capacity of a range of containers and recording measurements
- displacement activities to measure the volume of air or solids e.g., measuring the liquid displaced when a blown up balloon is put into a bucket of water
- creating booklets and tables
- applying knowledge to real life tasks e.g., cooking, science experiments
- research activities related to liquid capacity e.g., specialised units, units used in the past or in other countries.
Resources include:
- calibrated containers e.g., cups, jugs
- card material
- containers in environment
- command cards; word problems
- reference and research materials (paper-based, digital, web-based, multimedia).
＋ Links to Australian Curriculum (ACv9)
AC9M3M01
AC9M2M01
AC9M3M02
Length 2.GM.160
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand and apply the concept of length
.02 Learn units of measurement for length (metre, centimetre, millimetre, kilometre)
.03 Measure and record length of objects
.04 Measure and record heights (centimetres, metres)
.05 Research and record distances (kilometres)
.06 Choose the most appropriate unit of length depending on size of object, distance etc
.07 Research other measurements of length
v#### ＋ Materials and Activities
Activities include:
- activities to become familiar with units for measuring length including history of the metre, relationship between metre and smaller units (centimetre, millimetre) and larger units (kilometre), etymology and meaning of the word metre and prefixes (milli-, centi-, kilo-), abbreviations
- measuring the length of a range of objects, heights and distances and recording measurements
- researching and recording heights, distances of familiar journeys
- creating booklets and tables
- applying knowledge to real life tasks e.g., construction projects, planning journeys, classroom height chart, making clothes or costumes
- research activities related to length e.g., specialised units, units used in the past or in other countries, units used in sport.
Resources include:
- ruler, metre stick and tape measure
- card material
- objects and people in environment
- grid paper
- command cards; word problems
- reference and research materials (paper-based, digital, web-based, multimedia).
＋ Links to Australian Curriculum (ACv9)
AC9M1M02
AC9M3M01
AC9M2M01
AC9M3M02
Mass 2.GM.170
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand and apply the concept of mass
.02 Learn units of measurement for mass (gram, milligram, kilogram)
.03 Measure and record mass of objects
.04 Choose the most appropriate unit of mass depending on object
.05 Research other measurements of mass
＋ Materials and Activities
Activities include:
- activities to become familiar with the concept of mass e.g., using informal units to measure mass; hefting activities to compare and estimate the mass of objects
- activities to become familiar with standard units for measuring mass including history of the gram, relationship between gram and smaller units (milligram) and larger units (kilogram), etymology and meaning of the word gram and prefixes (milli-, kilo-), abbreviations
- measuring the mass of a range of objects, and recording measurements
- creating booklets and tables
- applying knowledge to real life tasks e.g., cooking, science experiments, health
- research activities related to mass e.g., specialised units, units used in the past or in other countries.
Resources include:
- ruler, metre stick and tape measure
- card material
- objects and people in environment
- grid paper
- command cards; word problems
- reference and research materials (paper-based, digital, web-based, multimedia).
＋ Links to Australian Curriculum (ACv9)
AC9M3M01
AC9M2M01
AC9M3M02
Temperature 2.GM.180
Knowledge, Skills and Understanding
Typically, children will:
.01 Understand and apply the concept of temperature
.02 Learn units of measurement for temperature (degrees)
.03 Measure and record temperature
.04 Research other measurements of temperature
＋ Materials and Activities
Activities include:
- using hands/fingertips to estimate and compare temperatures of familiar objects and materials
- activities to become familiar with units for measuring temperature (degrees) including history, etymology and abbreviations
- using a liquid thermometer to measure the temperature of liquids
- measuring the outside and inside temperature at different times of the day, in the sun and in the shade etc
- using a thermometer to measure own temperature or record body temperature changes over a day
- creating booklets and tables
- research activities related to temperature e.g., units used in the past or in other countries, meaning of changes in body temperature, meaning of average temperatures in climate science.
Resources include:
- different types of thermometers
- card material
- command cards; word problems
- reference and research materials (paper-based, digital, web-based, multimedia).
Estimation 2.GM.190
Knowledge, Skills and Understanding
Typically, children will:
.01 Estimate measurements of a range of phenomena
＋ Materials and Activities
Activities include:
- all introductory measurement activities
- activities involving estimation, comparison and graphing
- applying estimation skills to real life tasks e.g., in cooking, science experiments, planning going out activities and trips.
Resources include:
- all measurement materials
- command cards; word problems.
＋ Links to Australian Curriculum (ACv9)
AC9M3M01